Column Generation Based Primal Heuristics


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1 Column Generation Based Primal Heuristics C. Joncour, S. Michel, R. Sadykov, D. Sverdlov, F. Vanderbeck University Bordeaux 1 & INRIA team RealOpt
2 Outline 1 Context Generic Primal Heuristics The BranchandPrice Approach 2 Column Generation based Primal Heuristics Specificity of the BaP context Literature Review 3 Diving Heuristic An implementation with Limited Discrepancy Search Numerical tests 4 Discussion
3 Context Generic Primal Heuristics for MIPs good feasible solutions using the tools of exact optimization Truncating an exact method Building from the relaxation used for the exact approach Defining a target based on the relaxation Using dual information to price choices in greedy heuristics Exact approach used to explore a neighborhood Examples: [Berthold 06] 1 Relaxation Induced Neighborhood Search [Dana al 05] 2 Local Branching [Fischetti al 03] 3 Feasibility Pump [Fischetti al 05]
4 Context Generic Primal Heuristics for MIPs good feasible solutions using the tools of exact optimization Truncating an exact method Building from the relaxation used for the exact approach Defining a target based on the relaxation Using dual information to price choices in greedy heuristics Exact approach used to explore a neighborhood Examples: [Berthold 06] 1 Relaxation Induced Neighborhood Search [Dana al 05] 2 Local Branching [Fischetti al 03] 3 Feasibility Pump [Fischetti al 05]
5 Context Generic Primal Heuristics for MIPs good feasible solutions using the tools of exact optimization Truncating an exact method Building from the relaxation used for the exact approach Defining a target based on the relaxation Using dual information to price choices in greedy heuristics Exact approach used to explore a neighborhood Examples: [Berthold 06] 1 Relaxation Induced Neighborhood Search [Dana al 05] 2 Local Branching [Fischetti al 03] 3 Feasibility Pump [Fischetti al 05]
6 Context Generic Primal Heuristics for MIPs good feasible solutions using the tools of exact optimization Truncating an exact method Building from the relaxation used for the exact approach Defining a target based on the relaxation Using dual information to price choices in greedy heuristics Exact approach used to explore a neighborhood Examples: [Berthold 06] 1 Relaxation Induced Neighborhood Search [Dana al 05] 2 Local Branching [Fischetti al 03] 3 Feasibility Pump [Fischetti al 05]
7 Context Generic Primal Heuristics for MIPs good feasible solutions using the tools of exact optimization Truncating an exact method Building from the relaxation used for the exact approach Defining a target based on the relaxation Using dual information to price choices in greedy heuristics Exact approach used to explore a neighborhood Examples: [Berthold 06] 1 Relaxation Induced Neighborhood Search [Dana al 05] 2 Local Branching [Fischetti al 03] 3 Feasibility Pump [Fischetti al 05]
8 Context Generic LP based heuristics min{ j c jx j : j a ijx j b i i, l j x j u j j} Rounding: Iteratively select a var x j and bound/fix it least fractional: argmin j {min{x j x j, x j x j }} least infeasibility: argmin j {uplock j + downlock j } where uplock j = # constraint i with a ij < 0 and downlock j = # constraint i with a ij > 0 guided search: argmin j { x j x inc j } least pseudocost: argmin j {LP val increase} greedy: argmin j { c j Pi a } ij Diving: rounding + LP resolve + reiterate heuristic depth search in branchandbound tree branching rule that of exact branchandbound submiping: rounding + MIP sol of the residual prob.
9 Context Generic LP based heuristics min{ j c jx j : j a ijx j b i i, l j x j u j j} Rounding: Iteratively select a var x j and bound/fix it least fractional: argmin j {min{x j x j, x j x j }} least infeasibility: argmin j {uplock j + downlock j } where uplock j = # constraint i with a ij < 0 and downlock j = # constraint i with a ij > 0 guided search: argmin j { x j x inc j } least pseudocost: argmin j {LP val increase} greedy: argmin j { c j Pi a } ij Diving: rounding + LP resolve + reiterate heuristic depth search in branchandbound tree branching rule that of exact branchandbound submiping: rounding + MIP sol of the residual prob.
10 Context Generic LP based heuristics min{ j c jx j : j a ijx j b i i, l j x j u j j} Rounding: Iteratively select a var x j and bound/fix it least fractional: argmin j {min{x j x j, x j x j }} least infeasibility: argmin j {uplock j + downlock j } where uplock j = # constraint i with a ij < 0 and downlock j = # constraint i with a ij > 0 guided search: argmin j { x j x inc j } least pseudocost: argmin j {LP val increase} greedy: argmin j { c j Pi a } ij Diving: rounding + LP resolve + reiterate heuristic depth search in branchandbound tree branching rule that of exact branchandbound submiping: rounding + MIP sol of the residual prob.
11 Context Generic LP based heuristics min{ j c jx j : j a ijx j b i i, l j x j u j j} Rounding: Iteratively select a var x j and bound/fix it least fractional: argmin j {min{x j x j, x j x j }} least infeasibility: argmin j {uplock j + downlock j } where uplock j = # constraint i with a ij < 0 and downlock j = # constraint i with a ij > 0 guided search: argmin j { x j x inc j } least pseudocost: argmin j {LP val increase} greedy: argmin j { c j Pi a } ij Diving: rounding + LP resolve + reiterate heuristic depth search in branchandbound tree branching rule that of exact branchandbound submiping: rounding + MIP sol of the residual prob.
12 Context Heuristic search in branchandbound tree Diving submiping
13 Context The BranchandPrice Approach min c 1 x 1 + c 2 x c K x K D x 1 + D x D x K d B x 1 b B x 2 b B x K x 1 N n, x 2 N n,... x K N n. Relax Dx d = decomposition: subproblem {B x b, x N n } and a reformulation solved by BranchandPrice: b min P S S cx s λ s P s S Dx s λ s d P S S λs = K λ N S Solve Master LP Pricing Problem y := X k x k = X S S cx s λ s Solve Master LP Solve Master LP Pricing Problem Pricing Problem
14 Context The BranchandPrice Approach min c 1 x 1 + c 2 x c K x K D x 1 + D x D x K d B x 1 b B x 2 b B x K x 1 N n, x 2 N n,... x K N n. Relax Dx d = decomposition: subproblem {B x b, x N n } and a reformulation solved by BranchandPrice: b min P S S cx s λ s P s S Dx s λ s d P S S λs = K λ N S Solve Master LP Pricing Problem y := X k x k = X S S cx s λ s Solve Master LP Solve Master LP Pricing Problem Pricing Problem
15 Context The BranchandPrice Approach min c 1 x 1 + c 2 x c K x K D x 1 + D x D x K d B x 1 b B x 2 b B x K x 1 N n, x 2 N n,... x K N n. Relax Dx d = decomposition: subproblem {B x b, x N n } and a reformulation solved by BranchandPrice: b min P S S cx s λ s P s S Dx s λ s d P S S λs = K λ N S Solve Master LP Pricing Problem y := X k x k = X S S cx s λ s Solve Master LP Solve Master LP Pricing Problem Pricing Problem
16 Column Generation based Primal Heuristics Difficulties in a BaP context Heuristic paradigm in original space or the reformulation? On master variables: λ Cannot fix bounds (as in rounding) Cannot define constraints (as in local branching) On original variables: x Cannot grasp individual SP var. after aggregation in the common case of identical SPs Cannot modify the SP structure required by the oracle What can be done: define a partial solution iteratively updated (lb on λ) apply col. gen. to the residual prob. (prepr. = bd x, λ)
17 Column Generation based Primal Heuristics Difficulties in a BaP context Heuristic paradigm in original space or the reformulation? On master variables: λ Cannot fix bounds (as in rounding) Cannot define constraints (as in local branching) On original variables: x Cannot grasp individual SP var. after aggregation in the common case of identical SPs Cannot modify the SP structure required by the oracle What can be done: define a partial solution iteratively updated (lb on λ) apply col. gen. to the residual prob. (prepr. = bd x, λ)
18 Column Generation based Primal Heuristics Difficulties in a BaP context Heuristic paradigm in original space or the reformulation? On master variables: λ Cannot fix bounds (as in rounding) Cannot define constraints (as in local branching) On original variables: x Cannot grasp individual SP var. after aggregation in the common case of identical SPs Cannot modify the SP structure required by the oracle What can be done: define a partial solution iteratively updated (lb on λ) apply col. gen. to the residual prob. (prepr. = bd x, λ)
19 Column Generation based Primal Heuristics Difficulties in a BaP context Heuristic paradigm in original space or the reformulation? On master variables: λ Cannot fix bounds (as in rounding) Cannot define constraints (as in local branching) On original variables: x Cannot grasp individual SP var. after aggregation in the common case of identical SPs Cannot modify the SP structure required by the oracle What can be done: define a partial solution iteratively updated (lb on λ) apply col. gen. to the residual prob. (prepr. = bd x, λ)
20 Column Generation based Primal Heuristics Review of column generation based heuristics 1 Restricted Master min{ S c sλ s : s Dx s λ s d, s λ s = K, λ s N s S} restricted set build either heuristically / from master LP 2 Greedy iteratively select column with best pseudocost / redcost 3 Rounding round down master LP solution + iterative roundup complete heuristically 4 Local search delete columns from incumbent rebuild heuristically genericity & feasibility issue
21 Column Generation based Primal Heuristics Review of column generation based heuristics 1 Restricted Master min{ S c sλ s : s Dx s λ s d, s λ s = K, λ s N s S} restricted set build either heuristically / from master LP 2 Greedy iteratively select column with best pseudocost / redcost 3 Rounding round down master LP solution + iterative roundup complete heuristically 4 Local search delete columns from incumbent rebuild heuristically genericity & feasibility issue
22 Column Generation based Primal Heuristics Review of column generation based heuristics 1 Restricted Master min{ S c sλ s : s Dx s λ s d, s λ s = K, λ s N s S} restricted set build either heuristically / from master LP 2 Greedy iteratively select column with best pseudocost / redcost 3 Rounding round down master LP solution + iterative roundup complete heuristically 4 Local search delete columns from incumbent rebuild heuristically genericity & feasibility issue
23 Column Generation based Primal Heuristics Review of column generation based heuristics 1 Restricted Master min{ S c sλ s : s Dx s λ s d, s λ s = K, λ s N s S} restricted set build either heuristically / from master LP 2 Greedy iteratively select column with best pseudocost / redcost 3 Rounding round down master LP solution + iterative roundup complete heuristically 4 Local search delete columns from incumbent rebuild heuristically genericity & feasibility issue
24 Column Generation based Primal Heuristics Review of column generation based heuristics 1 Restricted Master min{ S c sλ s : s Dx s λ s d, s λ s = K, λ s N s S} restricted set build either heuristically / from master LP 2 Greedy iteratively select column with best pseudocost / redcost 3 Rounding round down master LP solution + iterative roundup complete heuristically 4 Local search delete columns from incumbent rebuild heuristically genericity & feasibility issue
25 Column Generation based Primal Heuristics Review of column generation based heuristics 1 Restricted Master (generic, often leads to infeasibility) min{ S c sλ s : s Dx s λ s d, s λ s = K, λ s N s S} restricted set build either heuristically / from master LP 2 Greedy (application specific) iteratively select column with best pseudocost / redcost 3 Rounding (application specific) round down master LP solution + iterative roundup complete heuristically 4 Local search (application specific) delete columns from incumbent rebuild heuristically genericity & feasibility issue
26 Diving Heuristic Diving Heuristic: generic algorithm Generic way to restore feasibility: further column generation Step 1: Solve the current master LP. Step 2: Select columns into the current solution at heuristically set values: λ s λ s Step 3: If the partial solution defines a complete primal solution, record this solution and goto Step 6. Step 4: Update the master (RHS) and the pricing problems (SP var. bounds). Step 5: If residual master problem is shown infeasible through preprocessing, goto Step 6. Else, return to Step 1. Step 6: + diversification procedure: limited backtracking
27 Diving Heuristic Diving Heuristic: generic algorithm Generic way to restore feasibility: further column generation Step 1: Solve the current master LP. Step 2: Select columns into the current solution at heuristically set values: λ s λ s Step 3: If the partial solution defines a complete primal solution, record this solution and goto Step 6. Step 4: Update the master (RHS) and the pricing problems (SP var. bounds). Step 5: If residual master problem is shown infeasible through preprocessing, goto Step 6. Else, return to Step 1. Step 6: Stop. + diversification procedure: limited backtracking
28 Diving Heuristic Diving Heuristic: generic algorithm Generic way to restore feasibility: further column generation Step 1: Solve the current master LP. Step 2: Select columns into the current solution at heuristically set values: λ s λ s Step 3: If the partial solution defines a complete primal solution, record this solution and goto Step 6. Step 4: Update the master (RHS) and the pricing problems (SP var. bounds). Step 5: If residual master problem is shown infeasible through preprocessing, goto Step 6. Else, return to Step 1. Step 6: Backtrack. + diversification procedure: limited backtracking
29 Diving Heuristic Diving plus diversification Depthfirst search: select column: λ s λ s update master and SP apply preprocessing Backtracking: Limited Discrepancy Search MaxDiscrepancy = 2, MaxDepth = 3
30 Diving Heuristic Numerical tests: Cutting Stock & Vertex Coloring pure BaP DH w LDS problem size #inst. #nodes time gap time CSP % 2 CSP % 14 VCP % 26
31 Diving Heuristic Numerical tests: Bin Packing With Conflicts conflict conflict conflict 20 inst. bap bap + diving w LDS class not opt. av. time not opt. av. time max. time t t t t u u u u
32 Diving Heuristic Numerical tests: Bin Packing With Conflicts conflict conflict conflict Tabu Search Diving H DH LDS class gap time gap time gap time t % % 1.3 0% 1.5 t % % 2.5 0% 5.9 t % % 5.1 0% 32.1 t % % % u % % 2.6 0% 3.3 u % % 5.4 0% 14.6 u % % % 74.8 u % % % 514.1
33 Diving Heuristic Numerical tests: Vehicle Routing Diving H problem size #inst. # unsolved gap time VRP An % 70
34 Discussion Feedback on experimentation Rounding/Diving based on fixing master var. λ works when 1 Sufficiently many columns in the solution: s λ s = K with K 1 2 Objective function is flat : many solutions of the same costs 3 Most of the combinatorial difficulty is in the subproblem
35 Discussion Further research Exact BranchandPrice with heuristic oracle First experiment on CSP with greedy knapsack solver Seems no good when most of the difficulty is in the SP SubMIPing Fixing original variables in aggregate space
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